Boundedness of the positive-speed control cost at the critical time

Determine whether the null-controllability control cost for the transport–diffusion equation with positive transport speed M>0 remains bounded as the viscosity parameter tends to zero at the critical time T=2L/M.

Background

For positive transport speed M>0, the paper proves that the uniform null-controllability cost diverges as the viscosity parameter tends to zero when TM/L<2, while it tends to zero when TM/L>2. Consequently, the uniform null-controllability threshold is T=2L/M. These results do not determine the behavior exactly at the threshold itself: boundedness of the cost at T=2L/M could hold even though the cost vanishes for every strictly larger time, or it could fail. The authors explicitly leave this endpoint question unresolved.